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Volume Integral

  1. Sep 19, 2015 #1

    Zondrina

    User Avatar
    Homework Helper

    Hi everyone.

    I've been curious about a particular symbol, but I've never seen it used or mentioned in any context. I don't really have much information about its usage, so I thought I would ask around and see if anyone knew about its application.

    Screen Shot 2015-09-19 at 3.10.24 PM.png

    I saw this symbol in Microsoft word.

    How do we interpret it, and how do we use it?

    I'm familiar with closed surface integrals with differential elements ##d \vec S##. We use those when we want to calculate the flux of a field ##\vec F##. I'm also familiar with closed surface integrals with differential elements ##dS##. We use those when we want to calculate surface area.

    What about the closed volume integral above though?

    I know we should probably use a differential element ##dV## for a closed volume, and the answer would represent the volume of the object. Is there such thing as a differential volume element ##d \vec V## such that we can extend theorems to the fourth dimension (theorem's like Stoke's theorem and the Divergence theorem)?

    Thank you in advance.
     
  2. jcsd
  3. Sep 19, 2015 #2

    jedishrfu

    Staff: Mentor

    I think the dash box is simply for inserting your integrand.

    Its up to you to remember the dV part.

    There is a seldom used math symbol called the delambertian thats used in relativity that is the 4D version of the del operator but this isnt it.
     
  4. Sep 19, 2015 #3

    Mark44

    Staff: Mentor

    Zondrina, I think you're asking about the integration symbol, not the box to the right. According to this page, https://en.wikipedia.org/wiki/Integral_symbol, that's a closed volume integral. I don't know much more about it, and a quick web search didn't turn up much.
     
  5. Sep 19, 2015 #4

    Zondrina

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    Homework Helper

    So the only reason the loop is around the triple integral is to signify the volume is closed.

    Does that mean something like the divergence theorem can be written like so:

    Screen Shot 2015-09-19 at 6.09.47 PM.png

    For a closed volume ##V## such as ##x^2 + y^2 + z^2 \leq 1##.

    For a volume ##V## that isn't closed such as ##x^2 + y^2 + z^2 < 1##, would the theorem would take the form:

    Screen Shot 2015-09-19 at 6.12.46 PM.png

    Otherwise I don't see any reason to ever have to use the symbol mentioned in the OP.
     
    Last edited: Sep 19, 2015
  6. Sep 20, 2015 #5
    I don't know what is meant by "closed volume". What is the difference between a closed volume and an open volume?
     
  7. Sep 20, 2015 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    In three dimensions there is no such thing as a "closed volume". There can be in higher dimensions, of course.
     
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